Question

find the area of triangle whose side are in the ratio 5: 12: 13: and its perimeter is 60cm​

Answer 1

Solution :-

Ratio of sides of a triangle = 5 : 12 : 13

Let the constant ratio be x

Sides of the triangle :

• a = 5x
• b = 12x
• c = 13x

Given perimeter = 60 cm

Semi perimeter of the triangle s = (a + b + c)/2

⇒ s = (5x + 12x + 13x)/2

⇒ s = 30x/2

⇒ s = 15x

Also, s = Perimeter/2 = 60/2 = 30 cm

⇒ s = 30

⇒ 15x = 30

⇒ x = 30/15 = 2

Sides of the triangle :

• a = 5x = 5 * 2 = 10 cm
• b = 12x = 12 * 2 = 24 cm
• c = 13x = 13 * 2 = 26 cm

Semi perimeter s = 15x = 15 * 2 = 30 cm

By using Heron's formula

Area of the triangle A = √[ s(s - a)(s - b)(s - c)

[ Where a,b,c are the sides of a triangle and s is semi perimeter ]

Substituting the value in the formula

⇒ A = √[ 30(30 - 10)(30 - 24)(30 - 26) ]

⇒ A = √[ 30(20)(6)(4) ]

⇒ A = √14400

⇒ A = 120 cm²

Hence, area of the triangle is 120 cm².

Answer 2

$\Large\underline{\underline{\sf{Given}:}}$

• Sides Ratio = 5 : 12 : 13 .
• Perimeter = 60cm .

$\Large\bold\star\underline{\underline\textbf{Formula\:used}}$

• Perimeter of ∆ = sum of all 3 sides.
• Area of ∆ = √s(s-a)(s-b)(s-c) where s = semi-perimeter , a,b and c are sides of ∆ .
• Pythagoras theoram.
• Area of Right angled ∆ = 1/2 × Base × Perpendicular .

$\large\star{\underline{\tt{\red{Answer}}}}\star$

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$\Large\bold\star\underline{\underline\textbf{Solution(1)}}$

Let sides of ∆ be = 5x , 12x and 13x Respectively .

Perimeter of ∆ = 5x + 12x + 13x = 30x .

Given,

30x = 60

→ x = 2 .

Hence, sides are ,

5×2 = 10cm, 12×2 = 24cm , 13×2 = 26cm...

Now,

semiperimeter(s) = 60/2 = 30cm.

Hence, Area of ∆ = √30(30-10)(30-24)(30-26)

→ Area of ∆ = √30*20*6*4

→ Area of ∆ = √5*6*5*4*6*4

→ Area of ∆ = 5*6*4 = 120cm² .. (Ans)

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$\Large\bold\star\underline{\underline\textbf{Solution(2)}}$

After Finding sides of ∆ , we can see if they are pythagorean Triplets or not ,

we know that,

According to pythagoras theoram sum of square of two sides is Equal to third side ...

→ (10)² + (24)² = (26)²

→ 100 + 576 = 676

→ 676 = 676 = Proved ..

Hence, it is a Right angled ∆ , with Hypotenuse 26cm.

so,

Area of Right angled ∆ = 1/2 × 10 × 24 = 120cm² ....

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$\large\underline\textbf{Hope it Helps You.}$